# Generalized inverse

(Redirected from Pseudo inverse)

In mathematics, and in particular, algebra, a generalized inverse of an element x is an element y that has some properties of an inverse element but not necessarily all of them. Generalized inverses can be defined in any mathematical structure that involves associative multiplication, that is, in a semigroup. This article describes generalized inverses of a matrix ${\displaystyle A}$.

Formally, given a matrix ${\displaystyle A\in \mathbb {R} ^{n\times m}}$ and a matrix ${\displaystyle A^{\mathrm {g} }\in \mathbb {R} ^{m\times n}}$, ${\displaystyle A^{\mathrm {g} }}$ is a generalized inverse of ${\displaystyle A}$ if it satisfies the condition ${\displaystyle AA^{\mathrm {g} }A=A}$.[1][2][3]

The purpose of constructing a generalized inverse of a matrix is to obtain a matrix that can serve as an inverse in some sense for a wider class of matrices than invertible matrices. A generalized inverse exists for an arbitrary matrix, and when a matrix has a regular inverse, this inverse is its unique generalized inverse.[4]

## Motivation

Consider the linear system

${\displaystyle Ax=y}$

where ${\displaystyle A}$ is an ${\displaystyle n\times m}$ matrix and ${\displaystyle y\in {\mathcal {R}}(A)}$, the column space of ${\displaystyle A}$. If ${\displaystyle A}$ is nonsingular, then ${\displaystyle x=A^{-1}y}$ will be the solution of the system. In this case, C. R. Rao and S. K. Mitra call ${\displaystyle A^{-1}}$ a regular inverse of ${\displaystyle A}$.[5] Note that, if ${\displaystyle A}$ is nonsingular, then

${\displaystyle AA^{-1}A=A.}$

Suppose ${\displaystyle A}$ is singular, or ${\displaystyle n\neq m}$. Then we need a right candidate ${\displaystyle G}$ of order ${\displaystyle m\times n}$ such that for all ${\displaystyle y\in {\mathcal {R}}(A)}$,

${\displaystyle AGy=y.}$[6]

That is, ${\displaystyle Gy}$ is a solution of the linear system ${\displaystyle Ax=y}$. Equivalently, we need a matrix ${\displaystyle G}$ of order ${\displaystyle m\times n}$ such that

${\displaystyle AGA=A.}$

Hence we can define the generalized inverse or g-inverse as follows: Given an ${\displaystyle n\times m}$ matrix ${\displaystyle A}$, an ${\displaystyle m\times n}$ matrix ${\displaystyle G}$ is said to be a generalized inverse of ${\displaystyle A}$ if ${\displaystyle AGA=A.}$[7][8][9]

## Types

The Penrose conditions define different generalized inverses for ${\displaystyle A\in \mathbb {R} ^{n\times m}}$ and ${\displaystyle A^{\mathrm {g} }\in \mathbb {R} ^{m\times n}:}$

1. ${\displaystyle AA^{\mathrm {g} }A=A}$
2. ${\displaystyle A^{\mathrm {g} }AA^{\mathrm {g} }=A^{\mathrm {g} }}$
3. ${\displaystyle (AA^{\mathrm {g} })^{\mathrm {T} }=AA^{\mathrm {g} }}$
4. ${\displaystyle (A^{\mathrm {g} }A)^{\mathrm {T} }=A^{\mathrm {g} }A,}$

where ${\displaystyle ^{\mathrm {T} }}$ indicates conjugate transpose. If ${\displaystyle A^{\mathrm {g} }}$ satisfies the first condition, then it is a generalized inverse of ${\displaystyle A}$. If it satisfies the first two conditions, then it is a reflexive generalized inverse of ${\displaystyle A}$. If it satisfies all four conditions, then it is a pseudoinverse of ${\displaystyle A}$.[10][11][12][13] A pseudoinverse is sometimes called the Moore–Penrose inverse, after the pioneering works by E. H. Moore and Roger Penrose.[14][15][16][17][18] When ${\displaystyle A}$ is non-singular, ${\displaystyle A^{\mathrm {g} }=A^{-1}}$ and is unique, but in all other cases, there is an infinite number of matrices that satisfy condition (1). However, the Moore–Penrose inverse is unique.[19]

There are other kinds of generalized inverse:

• One-sided inverse (right inverse or left inverse)
• Right inverse: If the matrix ${\displaystyle A}$ has dimensions ${\displaystyle n\times m}$ and ${\displaystyle {\textrm {rank}}(A)=n}$, then there exists an ${\displaystyle m\times n}$ matrix ${\displaystyle A_{\mathrm {R} }^{-1}}$ called the right inverse of ${\displaystyle A}$ such that ${\displaystyle AA_{\mathrm {R} }^{-1}=I_{n}}$, where ${\displaystyle I_{n}}$ is the ${\displaystyle n\times n}$ identity matrix.
• Left inverse: If the matrix ${\displaystyle A}$ has dimensions ${\displaystyle n\times m}$ and ${\displaystyle {\textrm {rank}}(A)=m}$, then there exists an ${\displaystyle m\times n}$ matrix ${\displaystyle A_{\mathrm {L} }^{-1}}$ called the left inverse of ${\displaystyle A}$ such that ${\displaystyle A_{\mathrm {L} }^{-1}A=I_{m}}$, where ${\displaystyle I_{m}}$ is the ${\displaystyle m\times m}$ identity matrix.[20]

## Examples

### Reflexive generalized inverse

Let

${\displaystyle A={\begin{bmatrix}1&2&3\\4&5&6\\7&8&9\end{bmatrix}},\quad G={\begin{bmatrix}-{\frac {5}{3}}&{\frac {2}{3}}&0\\[4pt]{\frac {4}{3}}&-{\frac {1}{3}}&0\\[4pt]0&0&0\end{bmatrix}}.}$

Since ${\displaystyle \det(A)=0}$, ${\displaystyle A}$ is singular and has no regular inverse. However, ${\displaystyle A}$ and ${\displaystyle G}$ satisfy conditions (1) and (2), but not (3) or (4). Hence, ${\displaystyle G}$ is a reflexive generalized inverse of ${\displaystyle A}$.

### One-sided inverse

Let

${\displaystyle A={\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}},\quad A_{\mathrm {R} }^{-1}={\begin{bmatrix}-{\frac {17}{18}}&{\frac {8}{18}}\\[4pt]-{\frac {2}{18}}&{\frac {2}{18}}\\[4pt]{\frac {13}{18}}&-{\frac {4}{18}}\end{bmatrix}}.}$

Since ${\displaystyle A}$ is not square, ${\displaystyle A}$ has no regular inverse. However, ${\displaystyle A_{\mathrm {R} }^{-1}}$ is a right inverse of ${\displaystyle A}$. The matrix ${\displaystyle A}$ has no left inverse

## Construction

The following characterizations are easy to verify:

1. A right inverse of a non-square matrix ${\displaystyle A}$ is given by ${\displaystyle A_{\mathrm {R} }^{-1}=A^{\mathrm {T} }\left(AA^{\mathrm {T} }\right)^{-1}}$.[21]
2. A left inverse of a non-square matrix ${\displaystyle A}$ is given by ${\displaystyle A_{\mathrm {L} }^{-1}=\left(A^{\mathrm {T} }A\right)^{-1}A^{\mathrm {T} }}$.[22]
3. If ${\displaystyle A=BC}$ is a rank factorization, then ${\displaystyle G=C_{\mathrm {R} }^{-1}B_{\mathrm {L} }^{-1}}$ is a g-inverse of ${\displaystyle A}$, where ${\displaystyle C_{\mathrm {R} }^{-1}}$ is a right inverse of ${\displaystyle C}$ and ${\displaystyle B_{\mathrm {L} }^{-1}}$ is left inverse of ${\displaystyle B}$.
4. If ${\displaystyle A=P{\begin{bmatrix}I_{r}&0\\0&0\end{bmatrix}}Q}$ for any non-singular matrices ${\displaystyle P}$ and ${\displaystyle Q}$, then ${\displaystyle G=Q^{-1}{\begin{bmatrix}I_{r}&U\\W&V\end{bmatrix}}P^{-1}}$ is a generalized inverse of ${\displaystyle A}$ for arbitrary ${\displaystyle U,V}$ and ${\displaystyle W}$.
5. Let ${\displaystyle A}$ be of rank ${\displaystyle r}$. Without loss of generality, let
${\displaystyle A={\begin{bmatrix}B&C\\D&E\end{bmatrix}},}$
where ${\displaystyle B_{r\times r}}$ is the non-singular submatrix of ${\displaystyle A}$. Then,
${\displaystyle G={\begin{bmatrix}B^{-1}&0\\0&0\end{bmatrix}}}$ is a g-inverse of ${\displaystyle A}$.

## Uses

Any generalized inverse can be used to determine whether a system of linear equations has any solutions, and if so to give all of them. If any solutions exist for the n × m linear system

${\displaystyle Ax=b}$,

with vector ${\displaystyle x}$ of unknowns and vector ${\displaystyle b}$ of constants, all solutions are given by

${\displaystyle x=A^{\mathrm {g} }b+[I-A^{\mathrm {g} }A]w}$,

parametric on the arbitrary vector ${\displaystyle w}$, where ${\displaystyle A^{\mathrm {g} }}$ is any generalized inverse of ${\displaystyle A}$. Solutions exist if and only if ${\displaystyle A^{\mathrm {g} }b}$ is a solution, that is, if and only if ${\displaystyle AA^{\mathrm {g} }b=b}$.[23]

## Notes

1. ^ Ben-Israel & Greville (2003, pp. 2,7)
2. ^ Nakamura (1991, pp. 41–42)
3. ^ Rao & Mitra (1971, pp. vii,20)
4. ^ Ben-Israel & Greville (2003, pp. 2,7)
5. ^ Rao & Mitra (1971, pp. 19–20)
6. ^ Rao & Mitra (1971, p. 24)
7. ^ Ben-Israel & Greville (2003, pp. 2,7)
8. ^ Nakamura (1991, pp. 41–42)
9. ^ Rao & Mitra (1971, pp. vii,20)
10. ^ Ben-Israel & Greville (2003, p. 7)
11. ^ Campbell & Meyer (1991, p. 9)
12. ^ Nakamura (1991, pp. 41–42)
13. ^ Rao & Mitra (1971, pp. 20,28,51)
14. ^ Ben-Israel & Greville (2003, p. 7)
15. ^ Campbell & Meyer (1991, p. 10)
16. ^ James (1978, p. 114)
17. ^ Nakamura (1991, p. 42)
18. ^ Rao & Mitra (1971, p. 50–51)
19. ^ James (1978, pp. 113–114)
20. ^ Rao & Mitra (1971, p. 19)
21. ^ Rao & Mitra (1971, p. 19)
22. ^ Rao & Mitra (1971, p. 19)
23. ^ James (1978, pp. 109–110)

## References

• Ben-Israel, Adi; Greville, Thomas N.E. (2003). Generalized inverses: Theory and applications (2nd ed.). New York, NY: Springer. ISBN 0-387-00293-6.
• Campbell, S. L.; Meyer, Jr., C. D. (1991). Generalized Inverses of Linear Transformations. Dover. ISBN 978-0-486-66693-8.
• James, M. (June 1978). "The generalised inverse". Mathematical Gazette. 62: 109–114. doi:10.2307/3617665.
• Nakamura, Yoshihiko (1991). Advanced Robotics: Redundancy and Optimization. Addison-Wesley. ISBN 0201151987.
• Rao, C. Radhakrishna; Mitra, Sujit Kumar (1971). Generalized Inverse of Matrices and its Applications. New York: John Wiley & Sons. p. 240. ISBN 0-471-70821-6.
• Zheng, B; Bapat, R. B. (2004). "Generalized inverse A(2)T,S and a rank equation". Applied Mathematics and Computation. 155: 407–415. doi:10.1016/S0096-3003(03)00786-0.